Question

warm up 30
the function $g(x) = -3x^2 + 9x + 12$ models the height (in meters) of a diver above the water $x$ seconds after diving off a platform.

1. find the x-intercepts of the function.
2. find the y-intercept of the function.
3. find the maximum or minimum value of the function.
4. explain the meaning of each key feature in the context of the diver’s motion.

Explanation:

1. Find the x - intercepts of the function.

Step 1: Set \(g(x)=0\)

We have the function \(g(x)=- 3x^{2}+9x + 12\). To find the x - intercepts, we set \(g(x) = 0\), so \(-3x^{2}+9x + 12=0\). We can divide the entire equation by \(-3\) to simplify it. Dividing each term by \(-3\) gives \(x^{2}-3x - 4=0\).

Step 2: Factor the quadratic equation

We factor the quadratic \(x^{2}-3x - 4\). We need two numbers that multiply to \(-4\) and add up to \(-3\). The numbers are \(-4\) and \(1\). So, \(x^{2}-3x - 4=(x - 4)(x+1)=0\).

Step 3: Solve for x

Using the zero - product property, if \((x - 4)(x + 1)=0\), then either \(x-4=0\) or \(x + 1=0\). Solving \(x-4=0\) gives \(x = 4\), and solving \(x + 1=0\) gives \(x=-1\). Since time \(x\) cannot be negative in this context (we are talking about time after diving off the platform), we consider \(x = 4\) as the relevant x - intercept for the context, but mathematically the x - intercepts are \(x=-1\) and \(x = 4\).

2. Find the y - intercept of the function.

Step 1: Set \(x = 0\)

To find the y - intercept of a function \(y = g(x)\), we set \(x = 0\) and evaluate \(g(0)\).

Step 2: Evaluate \(g(0)\)

Substitute \(x = 0\) into \(g(x)=-3x^{2}+9x + 12\). We get \(g(0)=-3(0)^{2}+9(0)+12=12\). So the y - intercept is \(y = 12\).

3. Find the maximum or minimum value of the function.

Step 1: Determine if it's a maximum or minimum

The function \(g(x)=-3x^{2}+9x + 12\) is a quadratic function in the form \(y=ax^{2}+bx + c\), where \(a=-3\), \(b = 9\), and \(c = 12\). Since \(a=-3<0\), the parabola opens downwards, so the function has a maximum value.

Step 2: Find the x - coordinate of the vertex

The x - coordinate of the vertex of a quadratic function \(y = ax^{2}+bx + c\) is given by \(x=-\frac{b}{2a}\). Substituting \(a=-3\) and \(b = 9\) into the formula, we get \(x=-\frac{9}{2\times(-3)}=\frac{9}{6}=\frac{3}{2}=1.5\).

Step 3: Find the maximum value

We substitute \(x = 1.5\) into the function \(g(x)\) to find the maximum value. \(g(1.5)=-3(1.5)^{2}+9(1.5)+12\). First, calculate \((1.5)^{2}=2.25\). Then, \(-3\times2.25=-6.75\), \(9\times1.5 = 13.5\). So, \(g(1.5)=-6.75 + 13.5+12=18.75\).

4. Explain the meaning of each key feature in the context of the diver’s motion.

Answer:

• x - intercepts: The x - intercepts are the values of \(x\) when \(g(x)=0\) (height above water is \(0\) meters). \(x=-1\) is not relevant in the context of the diver's motion (since time cannot be negative). \(x = 4\) represents the time (in seconds) when the diver hits the water.
• y - intercept: The y - intercept is the value of \(g(x)\) when \(x = 0\). When \(x = 0\), it is the time just as the diver dives off the platform. So, \(g(0)=12\) means that the height of the platform above the water is \(12\) meters.
• Maximum value: The maximum value of the function occurs at \(x = 1.5\) seconds and \(g(1.5)=18.75\) meters. This represents the maximum height the diver reaches above the water, \(1.5\) seconds after diving off the platform.

1. Answer (x - intercepts)

The x - intercepts of the function \(g(x)\) are \(x=-1\) and \(x = 4\) (in the context of the problem, the relevant one for the diver's motion is \(x = 4\) seconds).

2. Answer (y - intercept)

The y - intercept of the function \(g(x)\) is \(y = 12\) meters.

3. Answer (maximum value)

The maximum value of the function \(g(x)\) is \(18.75\) meters, which occurs at \(x = 1.5\) seconds.